Mercurial > hg > Members > kono > Proof > ZF-in-agda

annotate ordinal-definable.agda @ 94:4659a815b70d

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sat, 08 Jun 2019 13:18:10 +0900
parents d382a7902f5e
children f3da2c87cee0
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
rev   line source
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
1 open import Level
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
2 module ordinal-definable where
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
3
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
4 open import zf
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
5 open import ordinal
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
6
23
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 22
diff changeset
7 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ )
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
8
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
9 open import Relation.Binary.PropositionalEquality
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
10
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
11 open import Data.Nat.Properties
6
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
12 open import Data.Empty
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
13 open import Relation.Nullary
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
14
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
15 open import Relation.Binary
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
16 open import Relation.Binary.Core
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
17
27
bade0a35fdd9 OD, HOD, TC
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
18 -- Ordinal Definable Set
11
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 10
diff changeset
19
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
20 record OD {n : Level} : Set (suc n) where
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
21 field
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
22 def : (x : Ordinal {n} ) → Set n
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
23
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
24 open OD
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
25 open import Data.Unit
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
26
44
fcac01485f32 od→lv : {n : Level} → OD {n} → Nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 43
diff changeset
27 open Ordinal
fcac01485f32 od→lv : {n : Level} → OD {n} → Nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 43
diff changeset
28
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
29 postulate
45
33860eb44e47 od∅' {n} = ord→od (o∅ {n})
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 44
diff changeset
30 od→ord : {n : Level} → OD {n} → Ordinal {n}
36
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
31 ord→od : {n : Level} → Ordinal {n} → OD {n}
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
32
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
33 _∋_ : { n : Level } → ( a x : OD {n} ) → Set n
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
34 _∋_ {n} a x = def a ( od→ord x )
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
35
90
38d139b5edac def ord conversion
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 89
diff changeset
36 _c<_ : { n : Level } → ( x a : OD {n} ) → Set n
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
37 x c< a = a ∋ x
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
38
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
39 record _==_ {n : Level} ( a b : OD {n} ) : Set n where
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
40 field
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
41 eq→ : ∀ { x : Ordinal {n} } → def a x → def b x
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
42 eq← : ∀ { x : Ordinal {n} } → def b x → def a x
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
44 id : {n : Level} {A : Set n} → A → A
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
45 id x = x
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
46
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
47 eq-refl : {n : Level} { x : OD {n} } → x == x
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
48 eq-refl {n} {x} = record { eq→ = id ; eq← = id }
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
49
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
50 open _==_
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
51
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
52 eq-sym : {n : Level} { x y : OD {n} } → x == y → y == x
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
53 eq-sym eq = record { eq→ = eq← eq ; eq← = eq→ eq }
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
54
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
55 eq-trans : {n : Level} { x y z : OD {n} } → x == y → y == z → x == z
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
56 eq-trans x=y y=z = record { eq→ = λ t → eq→ y=z ( eq→ x=y t) ; eq← = λ t → eq← x=y ( eq← y=z t) }
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
57
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
58 _c≤_ : {n : Level} → OD {n} → OD {n} → Set (suc n)
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
59 a c≤ b = (a ≡ b) ∨ ( b ∋ a )
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
60
40
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
61 od∅ : {n : Level} → OD {n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
62 od∅ {n} = record { def = λ _ → Lift n ⊥ }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
63
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
64 postulate
36
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
65 c<→o< : {n : Level} {x y : OD {n} } → x c< y → od→ord x o< od→ord y
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
66 o<→c< : {n : Level} {x y : Ordinal {n} } → x o< y → ord→od x c< ord→od y
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
67 oiso : {n : Level} {x : OD {n}} → ord→od ( od→ord x ) ≡ x
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
68 diso : {n : Level} {x : Ordinal {n}} → od→ord ( ord→od x ) ≡ x
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
69 sup-od : {n : Level } → ( OD {n} → OD {n}) → OD {n}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
70 sup-c< : {n : Level } → ( ψ : OD {n} → OD {n}) → ∀ {x : OD {n}} → ψ x c< sup-od ψ
46
e584686a1307 == and ∅7
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 45
diff changeset
71
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
72 ∅1 : {n : Level} → ( x : OD {n} ) → ¬ ( x c< od∅ {n} )
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
73 ∅1 {n} x (lift ())
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
74
37
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
75 ∅3 : {n : Level} → { x : Ordinal {n}} → ( ∀(y : Ordinal {n}) → ¬ (y o< x ) ) → x ≡ o∅ {n}
81
96c932d0145d simpler ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 80
diff changeset
76 ∅3 {n} {x} = TransFinite {n} c2 c3 x where
30
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
77 c0 : Nat → Ordinal {n} → Set n
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
78 c0 lx x = (∀(y : Ordinal {n}) → ¬ (y o< x)) → x ≡ o∅ {n}
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
79 c2 : (lx : Nat) → c0 lx (record { lv = lx ; ord = Φ lx } )
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
80 c2 Zero not = refl
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
81 c2 (Suc lx) not with not ( record { lv = lx ; ord = Φ lx } )
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
82 ... | t with t (case1 ≤-refl )
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
83 c2 (Suc lx) not | t | ()
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
84 c3 : (lx : Nat) (x₁ : OrdinalD lx) → c0 lx (record { lv = lx ; ord = x₁ }) → c0 lx (record { lv = lx ; ord = OSuc lx x₁ })
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
85 c3 lx (Φ .lx) d not with not ( record { lv = lx ; ord = Φ lx } )
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
86 ... | t with t (case2 Φ< )
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
87 c3 lx (Φ .lx) d not | t | ()
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
88 c3 lx (OSuc .lx x₁) d not with not ( record { lv = lx ; ord = OSuc lx x₁ } )
34
c9ad0d97ce41 fix oridinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 33
diff changeset
89 ... | t with t (case2 (s< s<refl ) )
30
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
90 c3 lx (OSuc .lx x₁) d not | t | ()
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
91
36
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
92 def-subst : {n : Level } {Z : OD {n}} {X : Ordinal {n} }{z : OD {n}} {x : Ordinal {n} }→ def Z X → Z ≡ z → X ≡ x → def z x
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
93 def-subst df refl refl = df
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
94
69
93abc0133b8a union continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 68
diff changeset
95 transitive : {n : Level } { z y x : OD {suc n} } → z ∋ y → y ∋ x → z ∋ x
93abc0133b8a union continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 68
diff changeset
96 transitive {n} {z} {y} {x} z∋y x∋y with ordtrans ( c<→o< {suc n} {x} {y} x∋y ) ( c<→o< {suc n} {y} {z} z∋y )
36
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
97 ... | t = lemma0 (lemma t) where
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
98 lemma : ( od→ord x ) o< ( od→ord z ) → def ( ord→od ( od→ord z )) ( od→ord ( ord→od ( od→ord x )))
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
99 lemma xo<z = o<→c< xo<z
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
100 lemma0 : def ( ord→od ( od→ord z )) ( od→ord ( ord→od ( od→ord x ))) → def z (od→ord x)
69
93abc0133b8a union continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 68
diff changeset
101 lemma0 dz = def-subst {suc n} { ord→od ( od→ord z )} { od→ord ( ord→od ( od→ord x))} dz (oiso) (diso)
36
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
102
41
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
103 record Minimumo {n : Level } (x : Ordinal {n}) : Set (suc n) where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
104 field
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
105 mino : Ordinal {n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
106 min<x : mino o< x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
107
57
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
108 ∅5 : {n : Level} → { x : Ordinal {n} } → ¬ ( x ≡ o∅ {n} ) → o∅ {n} o< x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
109 ∅5 {n} {record { lv = Zero ; ord = (Φ .0) }} not = ⊥-elim (not refl)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
110 ∅5 {n} {record { lv = Zero ; ord = (OSuc .0 ord) }} not = case2 Φ<
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
111 ∅5 {n} {record { lv = (Suc lv) ; ord = ord }} not = case1 (s≤s z≤n)
37
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
112
46
e584686a1307 == and ∅7
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 45
diff changeset
113 ord-iso : {n : Level} {y : Ordinal {n} } → record { lv = lv (od→ord (ord→od y)) ; ord = ord (od→ord (ord→od y)) } ≡ record { lv = lv y ; ord = ord y }
e584686a1307 == and ∅7
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 45
diff changeset
114 ord-iso = cong ( λ k → record { lv = lv k ; ord = ord k } ) diso
44
fcac01485f32 od→lv : {n : Level} → OD {n} → Nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 43
diff changeset
115
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
116 -- avoiding lv != Zero error
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
117 orefl : {n : Level} → { x : OD {n} } → { y : Ordinal {n} } → od→ord x ≡ y → od→ord x ≡ y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
118 orefl refl = refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
119
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
120 ==-iso : {n : Level} → { x y : OD {n} } → ord→od (od→ord x) == ord→od (od→ord y) → x == y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
121 ==-iso {n} {x} {y} eq = record {
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
122 eq→ = λ d → lemma ( eq→ eq (def-subst d (sym oiso) refl )) ;
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
123 eq← = λ d → lemma ( eq← eq (def-subst d (sym oiso) refl )) }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
124 where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
125 lemma : {x : OD {n} } {z : Ordinal {n}} → def (ord→od (od→ord x)) z → def x z
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
126 lemma {x} {z} d = def-subst d oiso refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
127
57
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
128 =-iso : {n : Level } {x y : OD {suc n} } → (x == y) ≡ (ord→od (od→ord x) == y)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
129 =-iso {_} {_} {y} = cong ( λ k → k == y ) (sym oiso)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
130
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
131 ord→== : {n : Level} → { x y : OD {n} } → od→ord x ≡ od→ord y → x == y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
132 ord→== {n} {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
133 lemma : ( ox oy : Ordinal {n} ) → ox ≡ oy → (ord→od ox) == (ord→od oy)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
134 lemma ox ox refl = eq-refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
135
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
136 o≡→== : {n : Level} → { x y : Ordinal {n} } → x ≡ y → ord→od x == ord→od y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
137 o≡→== {n} {x} {.x} refl = eq-refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
138
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
139 >→¬< : {x y : Nat } → (x < y ) → ¬ ( y < x )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
140 >→¬< (s≤s x<y) (s≤s y<x) = >→¬< x<y y<x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
141
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
142 c≤-refl : {n : Level} → ( x : OD {n} ) → x c≤ x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
143 c≤-refl x = case1 refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
144
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
145 o<→o> : {n : Level} → { x y : OD {n} } → (x == y) → (od→ord x ) o< ( od→ord y) → ⊥
52
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 51
diff changeset
146 o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case1 lt) with
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 51
diff changeset
147 yx (def-subst {n} {ord→od (od→ord y)} {od→ord (ord→od (od→ord x))} (o<→c< (case1 lt )) oiso diso )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 51
diff changeset
148 ... | oyx with o<¬≡ (od→ord x) (od→ord x) refl (c<→o< oyx )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 51
diff changeset
149 ... | ()
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 51
diff changeset
150 o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case2 lt) with
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 51
diff changeset
151 yx (def-subst {n} {ord→od (od→ord y)} {od→ord (ord→od (od→ord x))} (o<→c< (case2 lt )) oiso diso )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 51
diff changeset
152 ... | oyx with o<¬≡ (od→ord x) (od→ord x) refl (c<→o< oyx )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 51
diff changeset
153 ... | ()
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 51
diff changeset
154
79
c07c548b2ac1 add some lemma
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
155 ==→o≡ : {n : Level} → { x y : Ordinal {suc n} } → ord→od x == ord→od y → x ≡ y
c07c548b2ac1 add some lemma
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
156 ==→o≡ {n} {x} {y} eq with trio< {n} x y
c07c548b2ac1 add some lemma
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
157 ==→o≡ {n} {x} {y} eq | tri< a ¬b ¬c = ⊥-elim ( o<→o> eq (o<-subst a (sym ord-iso) (sym ord-iso )))
c07c548b2ac1 add some lemma
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
158 ==→o≡ {n} {x} {y} eq | tri≈ ¬a b ¬c = b
c07c548b2ac1 add some lemma
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
159 ==→o≡ {n} {x} {y} eq | tri> ¬a ¬b c = ⊥-elim ( o<→o> (eq-sym eq) (o<-subst c (sym ord-iso) (sym ord-iso )))
c07c548b2ac1 add some lemma
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
160
90
38d139b5edac def ord conversion
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 89
diff changeset
161 ≡-def : {n : Level} → { x : Ordinal {suc n} } → x ≡ od→ord (record { def = λ z → z o< x } )
38d139b5edac def ord conversion
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 89
diff changeset
162 ≡-def {n} {x} = ==→o≡ {n} (subst (λ k → ord→od x == k ) (sym oiso) lemma ) where
38d139b5edac def ord conversion
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 89
diff changeset
163 lemma : ord→od x == record { def = λ z → z o< x }
38d139b5edac def ord conversion
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 89
diff changeset
164 eq→ lemma {w} z = subst₂ (λ k j → k o< j ) diso diso t where
38d139b5edac def ord conversion
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 89
diff changeset
165 t : (od→ord ( ord→od w)) o< (od→ord (ord→od x))
38d139b5edac def ord conversion
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 89
diff changeset
166 t = c<→o< {suc n} {ord→od w} {ord→od x} (def-subst {suc n} {_} {_} {ord→od x} {_} z refl (sym diso))
38d139b5edac def ord conversion
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 89
diff changeset
167 eq← lemma {w} z = def-subst {suc n} {_} {_} {ord→od x} {w} ( o<→c< {suc n} {_} {_} z ) refl diso
38d139b5edac def ord conversion
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 89
diff changeset
168
38d139b5edac def ord conversion
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 89
diff changeset
169 od≡-def : {n : Level} → { x : Ordinal {suc n} } → ord→od x ≡ record { def = λ z → z o< x }
38d139b5edac def ord conversion
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 89
diff changeset
170 od≡-def {n} {x} = subst (λ k → ord→od x ≡ k ) oiso (cong ( λ k → ord→od k ) (≡-def {n} {x} ))
38d139b5edac def ord conversion
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 89
diff changeset
171
91
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
172 ∋→o< : {n : Level} → { a x : OD {suc n} } → a ∋ x → od→ord x o< od→ord a
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
173 ∋→o< {n} {a} {x} lt = t where
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
174 t : (od→ord x) o< (od→ord a)
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
175 t = c<→o< {suc n} {x} {a} lt
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
176
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
177 o<∋→ : {n : Level} → { a x : OD {suc n} } → od→ord x o< od→ord a → a ∋ x
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
178 o<∋→ {n} {a} {x} lt = subst₂ (λ k j → def k j ) oiso diso t where
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
179 t : def (ord→od (od→ord a)) (od→ord (ord→od (od→ord x)))
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
180 t = o<→c< {suc n} {od→ord x} {od→ord a} lt
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
181
80
461690d60d07 remove ∅-base-def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
182 o∅≡od∅ : {n : Level} → ord→od (o∅ {suc n}) ≡ od∅ {suc n}
461690d60d07 remove ∅-base-def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
183 o∅≡od∅ {n} with trio< {n} (o∅ {suc n}) (od→ord (od∅ {suc n} ))
461690d60d07 remove ∅-base-def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
184 o∅≡od∅ {n} | tri< a ¬b ¬c = ⊥-elim (lemma a) where
461690d60d07 remove ∅-base-def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
185 lemma : o∅ {suc n } o< (od→ord (od∅ {suc n} )) → ⊥
461690d60d07 remove ∅-base-def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
186 lemma lt with def-subst (o<→c< lt) oiso refl
461690d60d07 remove ∅-base-def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
187 lemma lt | lift ()
461690d60d07 remove ∅-base-def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
188 o∅≡od∅ {n} | tri≈ ¬a b ¬c = trans (cong (λ k → ord→od k ) b ) oiso
461690d60d07 remove ∅-base-def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
189 o∅≡od∅ {n} | tri> ¬a ¬b c = ⊥-elim (¬x<0 c)
461690d60d07 remove ∅-base-def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
190
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
191 o<→¬== : {n : Level} → { x y : OD {n} } → (od→ord x ) o< ( od→ord y) → ¬ (x == y )
52
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 51
diff changeset
192 o<→¬== {n} {x} {y} lt eq = o<→o> eq lt
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
193
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
194 o<→¬c> : {n : Level} → { x y : OD {n} } → (od→ord x ) o< ( od→ord y) → ¬ (y c< x )
81
96c932d0145d simpler ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 80
diff changeset
195 o<→¬c> {n} {x} {y} olt clt = o<> olt (c<→o< clt ) where
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
196
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
197 o≡→¬c< : {n : Level} → { x y : OD {n} } → (od→ord x ) ≡ ( od→ord y) → ¬ x c< y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
198 o≡→¬c< {n} {x} {y} oeq lt = o<¬≡ (od→ord x) (od→ord y) (orefl oeq ) (c<→o< lt)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
199
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
200 tri-c< : {n : Level} → Trichotomous _==_ (_c<_ {suc n})
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
201 tri-c< {n} x y with trio< {n} (od→ord x) (od→ord y)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
202 tri-c< {n} x y | tri< a ¬b ¬c = tri< (def-subst (o<→c< a) oiso diso) (o<→¬== a) ( o<→¬c> a )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
203 tri-c< {n} x y | tri≈ ¬a b ¬c = tri≈ (o≡→¬c< b) (ord→== b) (o≡→¬c< (sym b))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
204 tri-c< {n} x y | tri> ¬a ¬b c = tri> ( o<→¬c> c) (λ eq → o<→¬== c (eq-sym eq ) ) (def-subst (o<→c< c) oiso diso)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
205
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
206 c<> : {n : Level } { x y : OD {suc n}} → x c< y → y c< x → ⊥
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
207 c<> {n} {x} {y} x<y y<x with tri-c< x y
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
208 c<> {n} {x} {y} x<y y<x | tri< a ¬b ¬c = ¬c y<x
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
209 c<> {n} {x} {y} x<y y<x | tri≈ ¬a b ¬c = o<→o> b ( c<→o< x<y )
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
210 c<> {n} {x} {y} x<y y<x | tri> ¬a ¬b c = ¬a x<y
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
211
60
6a1f67a4cc6e dead end
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 59
diff changeset
212 ∅< : {n : Level} → { x y : OD {n} } → def x (od→ord y ) → ¬ ( x == od∅ {n} )
6a1f67a4cc6e dead end
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 59
diff changeset
213 ∅< {n} {x} {y} d eq with eq→ eq d
6a1f67a4cc6e dead end
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 59
diff changeset
214 ∅< {n} {x} {y} d eq | lift ()
57
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
215
78
9a7a64b2388c infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
216 ∅6 : {n : Level} → { x : OD {suc n} } → ¬ ( x ∋ x ) -- no Russel paradox
9a7a64b2388c infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
217 ∅6 {n} {x} x∋x = c<> {n} {x} {x} x∋x x∋x
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
218
76
8e8f54e7a030 extensionality done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
219 def-iso : {n : Level} {A B : OD {n}} {x y : Ordinal {n}} → x ≡ y → (def A y → def B y) → def A x → def B x
8e8f54e7a030 extensionality done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
220 def-iso refl t = t
8e8f54e7a030 extensionality done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
221
53
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 52
diff changeset
222 is-∋ : {n : Level} → ( x y : OD {suc n} ) → Dec ( x ∋ y )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 52
diff changeset
223 is-∋ {n} x y with tri-c< x y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 52
diff changeset
224 is-∋ {n} x y | tri< a ¬b ¬c = no ¬c
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 52
diff changeset
225 is-∋ {n} x y | tri≈ ¬a b ¬c = no ¬c
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 52
diff changeset
226 is-∋ {n} x y | tri> ¬a ¬b c = yes c
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 52
diff changeset
227
57
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
228 is-o∅ : {n : Level} → ( x : Ordinal {suc n} ) → Dec ( x ≡ o∅ {suc n} )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
229 is-o∅ {n} record { lv = Zero ; ord = (Φ .0) } = yes refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
230 is-o∅ {n} record { lv = Zero ; ord = (OSuc .0 ord₁) } = no ( λ () )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
231 is-o∅ {n} record { lv = (Suc lv₁) ; ord = ord } = no (λ())
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
232
45
33860eb44e47 od∅' {n} = ord→od (o∅ {n})
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 44
diff changeset
233 open _∧_
33860eb44e47 od∅' {n} = ord→od (o∅ {n})
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 44
diff changeset
234
66
92a11dc6425c regularity done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 65
diff changeset
235 ¬∅=→∅∈ : {n : Level} → { x : OD {suc n} } → ¬ ( x == od∅ {suc n} ) → x ∋ od∅ {suc n}
92a11dc6425c regularity done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 65
diff changeset
236 ¬∅=→∅∈ {n} {x} ne = def-subst (lemma (od→ord x) (subst (λ k → ¬ (k == od∅ {suc n} )) (sym oiso) ne )) oiso refl where
92a11dc6425c regularity done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 65
diff changeset
237 lemma : (ox : Ordinal {suc n}) → ¬ (ord→od ox == od∅ {suc n} ) → ord→od ox ∋ od∅ {suc n}
92a11dc6425c regularity done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 65
diff changeset
238 lemma ox ne with is-o∅ ox
92a11dc6425c regularity done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 65
diff changeset
239 lemma ox ne | yes refl with ne ( ord→== lemma1 ) where
92a11dc6425c regularity done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 65
diff changeset
240 lemma1 : od→ord (ord→od o∅) ≡ od→ord od∅
80
461690d60d07 remove ∅-base-def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
241 lemma1 = cong ( λ k → od→ord k ) o∅≡od∅
66
92a11dc6425c regularity done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 65
diff changeset
242 lemma o∅ ne | yes refl | ()
80
461690d60d07 remove ∅-base-def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
243 lemma ox ne | no ¬p = subst ( λ k → def (ord→od ox) (od→ord k) ) o∅≡od∅ (o<→c< (∅5 ¬p))
69
93abc0133b8a union continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 68
diff changeset
244
79
c07c548b2ac1 add some lemma
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
245 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
94
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 93
diff changeset
246 -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality (suc n) (suc (suc n))
59
d13d1351a1fa lemma = cong₂ (λ x not → minimul x not ) oiso { }6
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 58
diff changeset
247
94
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 93
diff changeset
248 Def : OD {suc n} → OD {suc n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 93
diff changeset
249 Def X = record { def = λ y → ∀ (x : Ordinal {suc n} ) → def X x → def (ord→od y) x }
89
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
250
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
251
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
252 OD→ZF : {n : Level} → ZF {suc (suc n)} {suc n}
40
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
253 OD→ZF {n} = record {
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
254 ZFSet = OD {suc n}
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
255 ; _∋_ = _∋_
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
256 ; _≈_ = _==_
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
257 ; ∅ = od∅
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
258 ; _,_ = _,_
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
259 ; Union = Union
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
260 ; Power = Power
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
261 ; Select = Select
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
262 ; Replace = Replace
81
96c932d0145d simpler ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 80
diff changeset
263 ; infinite = ord→od ( record { lv = Suc Zero ; ord = Φ 1 } )
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
264 ; isZF = isZF
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
265 } where
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
266 Replace : OD {suc n} → (OD {suc n} → OD {suc n} ) → OD {suc n}
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
267 Replace X ψ = sup-od ψ
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
268 Select : OD {suc n} → (OD {suc n} → Set (suc n) ) → OD {suc n}
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
269 Select X ψ = record { def = λ x → ( def X x ∧ ψ ( ord→od x )) }
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
270 _,_ : OD {suc n} → OD {suc n} → OD {suc n}
84
4b2aff372b7c omax ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 83
diff changeset
271 x , y = record { def = λ z → z o< (omax (od→ord x) (od→ord y)) }
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
272 Union : OD {suc n} → OD {suc n}
71
d088eb66564e add osuc ( next larger element of Ordinal )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 70
diff changeset
273 Union U = record { def = λ y → osuc y o< (od→ord U) }
77
75ba8cf64707 Power Set on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 76
diff changeset
274 -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ ( ∀ {x} → t ∋ x → X ∋ x )
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
275 Power : OD {suc n} → OD {suc n}
94
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 93
diff changeset
276 Power X = record { def = λ y → ∀ (x : Ordinal {suc n} ) → def (ord→od y) x → def X x }
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
277 ZFSet = OD {suc n}
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
278 _∈_ : ( A B : ZFSet ) → Set (suc n)
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
279 A ∈ B = B ∋ A
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
280 _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → Set (suc n)
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
281 _⊆_ A B {x} = A ∋ x → B ∋ x
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
282 _∩_ : ( A B : ZFSet ) → ZFSet
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
283 A ∩ B = Select (A , B) ( λ x → ( A ∋ x ) ∧ (B ∋ x) )
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
284 _∪_ : ( A B : ZFSet ) → ZFSet
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
285 A ∪ B = Select (A , B) ( λ x → (A ∋ x) ∨ ( B ∋ x ) )
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
286 infixr 200 _∈_
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
287 infixr 230 _∩_ _∪_
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
288 infixr 220 _⊆_
81
96c932d0145d simpler ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 80
diff changeset
289 isZF : IsZF (OD {suc n}) _∋_ _==_ od∅ _,_ Union Power Select Replace (ord→od ( record { lv = Suc Zero ; ord = Φ 1} ))
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
290 isZF = record {
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
291 isEquivalence = record { refl = eq-refl ; sym = eq-sym; trans = eq-trans }
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
292 ; pair = pair
72
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
293 ; union-u = union-u
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
294 ; union→ = union→
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
295 ; union← = union←
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
296 ; empty = empty
76
8e8f54e7a030 extensionality done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
297 ; power→ = power→
8e8f54e7a030 extensionality done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
298 ; power← = power←
8e8f54e7a030 extensionality done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
299 ; extensionality = extensionality
30
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
300 ; minimul = minimul
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
301 ; regularity = regularity
78
9a7a64b2388c infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
302 ; infinity∅ = infinity∅
93
d382a7902f5e replacement
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 91
diff changeset
303 ; infinity = λ _ → infinity
55
9c0a5e28a572 regurality elimination case
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 54
diff changeset
304 ; selection = λ {ψ} {X} {y} → selection {ψ} {X} {y}
93
d382a7902f5e replacement
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 91
diff changeset
305 ; replacement = replacement
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
306 } where
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
307 open _∧_
41
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
308 open Minimumo
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
309 pair : (A B : OD {suc n} ) → ((A , B) ∋ A) ∧ ((A , B) ∋ B)
87
296388c03358 split omax?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 84
diff changeset
310 proj1 (pair A B ) = omax-x {n} (od→ord A) (od→ord B)
296388c03358 split omax?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 84
diff changeset
311 proj2 (pair A B ) = omax-y {n} (od→ord A) (od→ord B)
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
312 empty : (x : OD {suc n} ) → ¬ (od∅ ∋ x)
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
313 empty x ()
94
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 93
diff changeset
314 --- Power X = record { def = λ t → ∀ (x : Ordinal {suc n} ) → def (ord→od t) x → def X x }
76
8e8f54e7a030 extensionality done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
315 power→ : (A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → A ∋ x
94
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 93
diff changeset
316 power→ A t P∋t {x} t∋x = ?
77
75ba8cf64707 Power Set on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 76
diff changeset
317 power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t
94
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 93
diff changeset
318 power← A t t→A z _ = ?
70
cd9cf8b09610 Union needs +1 space
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
319 union-u : (X z : OD {suc n}) → Union X ∋ z → OD {suc n}
72
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
320 union-u X z U>z = ord→od ( osuc ( od→ord z ) )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
321 union-lemma-u : {X z : OD {suc n}} → (U>z : Union X ∋ z ) → union-u X z U>z ∋ z
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
322 union-lemma-u {X} {z} U>z = lemma <-osuc where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
323 lemma : {oz ooz : Ordinal {suc n}} → oz o< ooz → def (ord→od ooz) oz
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
324 lemma {oz} {ooz} lt = def-subst {suc n} {ord→od ooz} (o<→c< lt) refl diso
73
dd430a95610f fix ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
325 union→ : (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
72
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
326 union→ X y u xx with trio< ( od→ord u ) ( osuc ( od→ord y ))
74
819da8c08f05 ordinal atomical successor?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
327 union→ X y u xx | tri< a ¬b ¬c with osuc-< a (c<→o< (proj2 xx))
819da8c08f05 ordinal atomical successor?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
328 union→ X y u xx | tri< a ¬b ¬c | ()
73
dd430a95610f fix ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
329 union→ X y u xx | tri≈ ¬a b ¬c = lemma b (c<→o< (proj1 xx )) where
dd430a95610f fix ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
330 lemma : {oX ou ooy : Ordinal {suc n}} → ou ≡ ooy → ou o< oX → ooy o< oX
dd430a95610f fix ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
331 lemma refl lt = lt
dd430a95610f fix ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
332 union→ X y u xx | tri> ¬a ¬b c = ordtrans {suc n} {osuc ( od→ord y )} {od→ord u} {od→ord X} c ( c<→o< (proj1 xx ))
72
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
333 union← : (X z : OD) (X∋z : Union X ∋ z) → (X ∋ union-u X z X∋z) ∧ (union-u X z X∋z ∋ z )
73
dd430a95610f fix ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
334 union← X z X∋z = record { proj1 = def-subst {suc n} (o<→c< X∋z) oiso refl ; proj2 = union-lemma-u X∋z }
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
335 ψiso : {ψ : OD {suc n} → Set (suc n)} {x y : OD {suc n}} → ψ x → x ≡ y → ψ y
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
336 ψiso {ψ} t refl = t
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
337 selection : {ψ : OD → Set (suc n)} {X y : OD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y)
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
338 selection {ψ} {X} {y} = record {
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
339 proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) }
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
340 ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso }
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
341 }
93
d382a7902f5e replacement
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 91
diff changeset
342 replacement : {ψ : OD → OD} (X x : OD) → Replace X ψ ∋ ψ x
d382a7902f5e replacement
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 91
diff changeset
343 replacement {ψ} X x = sup-c< ψ {x}
60
6a1f67a4cc6e dead end
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 59
diff changeset
344 ∅-iso : {x : OD} → ¬ (x == od∅) → ¬ ((ord→od (od→ord x)) == od∅)
6a1f67a4cc6e dead end
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 59
diff changeset
345 ∅-iso {x} neq = subst (λ k → ¬ k) (=-iso {n} ) neq
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
346 minimul : (x : OD {suc n} ) → ¬ (x == od∅ )→ OD {suc n}
68
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 66
diff changeset
347 minimul x not = od∅
57
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
348 regularity : (x : OD) (not : ¬ (x == od∅)) →
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
349 (x ∋ minimul x not) ∧ (Select (minimul x not) (λ x₁ → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅)
66
92a11dc6425c regularity done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 65
diff changeset
350 proj1 (regularity x not ) = ¬∅=→∅∈ not
92a11dc6425c regularity done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 65
diff changeset
351 proj2 (regularity x not ) = record { eq→ = reg ; eq← = λ () } where
92a11dc6425c regularity done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 65
diff changeset
352 reg : {y : Ordinal} → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) y → def od∅ y
92a11dc6425c regularity done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 65
diff changeset
353 reg {y} t with proj1 t
92a11dc6425c regularity done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 65
diff changeset
354 ... | x∈∅ = x∈∅
76
8e8f54e7a030 extensionality done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
355 extensionality : {A B : OD {suc n}} → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B
8e8f54e7a030 extensionality done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
356 eq→ (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {A} {B} (sym diso) (proj1 (eq (ord→od x))) d
8e8f54e7a030 extensionality done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
357 eq← (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {B} {A} (sym diso) (proj2 (eq (ord→od x))) d
89
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
358 xx-union : {x : OD {suc n}} → (x , x) ≡ record { def = λ z → z o< osuc (od→ord x) }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
359 xx-union {x} = cong ( λ k → record { def = λ z → z o< k } ) (omxx (od→ord x))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
360 xxx-union : {x : OD {suc n}} → (x , (x , x)) ≡ record { def = λ z → z o< osuc (osuc (od→ord x))}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
361 xxx-union {x} = cong ( λ k → record { def = λ z → z o< k } ) lemma where
91
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
362 lemma1 : {x : OD {suc n}} → od→ord x o< od→ord (x , x)
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
363 lemma1 {x} = c<→o< ( proj1 (pair x x ) )
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
364 lemma2 : {x : OD {suc n}} → od→ord (x , x) ≡ osuc (od→ord x)
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
365 lemma2 = trans ( cong ( λ k → od→ord k ) xx-union ) (sym ≡-def)
89
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
366 lemma : {x : OD {suc n}} → omax (od→ord x) (od→ord (x , x)) ≡ osuc (osuc (od→ord x))
91
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
367 lemma {x} = trans ( sym ( omax< (od→ord x) (od→ord (x , x)) lemma1 ) ) ( cong ( λ k → osuc k ) lemma2 )
90
38d139b5edac def ord conversion
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 89
diff changeset
368 uxxx-union : {x : OD {suc n}} → Union (x , (x , x)) ≡ record { def = λ z → osuc z o< osuc (osuc (od→ord x)) }
38d139b5edac def ord conversion
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 89
diff changeset
369 uxxx-union {x} = cong ( λ k → record { def = λ z → osuc z o< k } ) lemma where
38d139b5edac def ord conversion
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 89
diff changeset
370 lemma : od→ord (x , (x , x)) ≡ osuc (osuc (od→ord x))
91
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
371 lemma = trans ( cong ( λ k → od→ord k ) xxx-union ) (sym ≡-def )
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
372 uxxx-2 : {x : OD {suc n}} → record { def = λ z → osuc z o< osuc (osuc (od→ord x)) } == record { def = λ z → z o< osuc (od→ord x) }
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
373 eq→ ( uxxx-2 {x} ) {m} lt = proj1 (osuc2 m (od→ord x)) lt
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
374 eq← ( uxxx-2 {x} ) {m} lt = proj2 (osuc2 m (od→ord x)) lt
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
375 uxxx-ord : {x : OD {suc n}} → od→ord (Union (x , (x , x))) ≡ osuc (od→ord x)
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
376 uxxx-ord {x} = trans (cong (λ k → od→ord k ) uxxx-union) (==→o≡ (subst₂ (λ j k → j == k ) (sym oiso) (sym od≡-def ) uxxx-2 ))
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
377 omega = record { lv = Suc Zero ; ord = Φ 1 }
78
9a7a64b2388c infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
378 infinite : OD {suc n}
91
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
379 infinite = ord→od ( omega )
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
380 infinity∅ : ord→od ( omega ) ∋ od∅ {suc n}
78
9a7a64b2388c infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
381 infinity∅ = def-subst {suc n} {_} {od→ord (ord→od o∅)} {infinite} {od→ord od∅}
80
461690d60d07 remove ∅-base-def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
382 (o<→c< ( case1 (s≤s z≤n ))) refl (cong (λ k → od→ord k) o∅≡od∅ )
91
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
383 infinite∋x : (x : OD) → infinite ∋ x → od→ord x o< omega
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
384 infinite∋x x lt = subst (λ k → od→ord x o< k ) diso t where
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
385 t : od→ord x o< od→ord (ord→od (omega))
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
386 t = ∋→o< {n} {infinite} {x} lt
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
387 infinite∋uxxx : (x : OD) → osuc (od→ord x) o< omega → infinite ∋ Union (x , (x , x ))
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
388 infinite∋uxxx x lt = o<∋→ t where
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
389 t : od→ord (Union (x , (x , x))) o< od→ord (ord→od (omega))
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
390 t = subst (λ k → od→ord (Union (x , (x , x))) o< k ) (sym diso ) ( subst ( λ k → k o< omega ) ( sym (uxxx-ord {x} ) ) lt )
78
9a7a64b2388c infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
391 infinity : (x : OD) → infinite ∋ x → infinite ∋ Union (x , (x , x ))
91
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
392 infinity x lt = infinite∋uxxx x ( lemma (od→ord x) (infinite∋x x lt )) where
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
393 lemma : (ox : Ordinal {suc n} ) → ox o< omega → osuc ox o< omega
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
394 lemma record { lv = Zero ; ord = (Φ .0) } (case1 (s≤s x)) = case1 (s≤s z≤n)
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
395 lemma record { lv = Zero ; ord = (OSuc .0 ord₁) } (case1 (s≤s x)) = case1 (s≤s z≤n)
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
396 lemma record { lv = (Suc lv₁) ; ord = (Φ .(Suc lv₁)) } (case1 (s≤s ()))
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
397 lemma record { lv = (Suc lv₁) ; ord = (OSuc .(Suc lv₁) ord₁) } (case1 (s≤s ()))
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
398 lemma record { lv = 1 ; ord = (Φ 1) } (case2 c2) with d<→lv c2
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
399 lemma record { lv = (Suc Zero) ; ord = (Φ .1) } (case2 ()) | refl
78
9a7a64b2388c infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
400